CLAIM C3 of entrylib:2026.000002 Admissibility of lagged state space models through the ARIMA form of the whole model
licence CC-BY-4.0; contributions C3

DEFINITIONS
D1 [method] lagged SSOE model: A pure additive single-source-of-error state space model with $k$ components, $$\begin{aligned} y_t &= w' v_{t-l} + \epsilon_t \\ v_t &= F v_{t-l} + g \epsilon_t, \end{aligned}$$ in which component $j$ of $v_{t-l}$ enters with its own lag $l_j$; its discount matrix is $D = F - g w'$ and its lag matrix is $L(B) = \mathrm{diag}(B^{l_1}, \dots, B^{l_k})$ for the backshift operator $B$; $\Lambda(\lambda) = \mathrm{diag}(\lambda^{l_1}, \dots, \lambda^{l_k})$ for a complex number $\lambda$.
D2 [concept] expanded discount matrix: The transition matrix $\tilde D$ of size $\sum_j l_j$ of the lag-1 system equivalent to $v_t = D v_{t-l}$, in which component $j$ carries itself and its $l_j - 1$ lagged copies; $\tilde F$, $\tilde g$ and $\tilde w$ are expanded in the same way, so that $\tilde D = \tilde F - \tilde g \tilde w'$.
D3 [concept] decaying weights: A lagged SSOE model has decaying weights when the weights $\tilde w' \tilde D^j \tilde g$ of the past observations in its fitted values tend to zero, that is, when every eigenvalue $\lambda$ of its expanded discount matrix that is reachable from $\tilde g$ and observable through $\tilde w$ has $|\lambda| < 1$; the largest such $|\lambda|$ is its largest mode modulus. A mode that $\tilde g$ does not reach but $\tilde w$ sees carries an initial state whose effect is not covered: with $|\lambda| > 1$ it grows.
D4 [method] per-lag check: The test applied by smoothEigens() of smooth 4.6.0 and of its repository until commit 4a44c1c (src/headers/eigenCalc.h) to models without adaptive regressors: every eigenvalue of every diagonal block of $D$, one block per unique lag, has modulus below 1; reimplemented in numpy as per_lag_max().
D5 [method] PBH reference: The largest modulus among the eigenvalues $\lambda$ of the expanded discount matrix for which the smallest singular values of $[\lambda I - \tilde D;\ \tilde w']$ and of $[\lambda I - \tilde D,\ \tilde g]$ both exceed $10^{-7}(\|\tilde D\|_2 + \|\tilde g\| + \|\tilde w\|)$ (Popov-Belevitch-Hautus rank tests), computed with numpy as pbh_max().
D6 [method] cyclotomic cancellation: The procedure that computes the coefficients of $$\begin{aligned} \theta(B) &= \det(I - D L(B)) \\ \phi(B) &= \det(I - F L(B)) \end{aligned}$$ by an inverse discrete Fourier transform of their values at the $N$-th roots of unity ($N$ the smallest power of 2 above $\sum_j l_j$), and divides both by every cyclotomic polynomial $\Phi_n(B)$, $n$ dividing one of the lags, for as long as it divides both with a remainder at most $10^{-9}$ times the sum of absolute coefficients; the quotient of $\theta$ is the cancelled MA polynomial $\theta^*(B)$.
D7 [method] step-down test: The Levinson-Durbin step-down (Schur-Cohn) recursion on $1 + c_1 B + \dots + c_n B^n$: for $j = n, \dots, 1$, $$\kappa_j = c_j, \qquad c_i \leftarrow \frac{c_i - \kappa_j c_{j-i}}{1 - \kappa_j^2};$$ the test passes when every $|\kappa_j| < 1$, which holds if and only if all roots lie outside the unit circle (the $\kappa_j$ are the partial autocorrelations of the autoregression with this polynomial). [source B2]
D8 [method] GUM: Generalised Univariate Model of the smooth package: a lagged SSOE model whose $F$, $g$ and $w$ are all estimated. [source B3]
D9 [method] ETS(A,N,A): Exponential smoothing with additive error, no trend and additive seasonality of period $m$; here in the lagged form with a level of lag 1 and one seasonal state of lag $m$, $F = I$, $w = (1, 1)'$, $g = (\alpha, \gamma)'$. [source B1]
D10 [method] check of the whole model: Admissibility of a lagged SSOE model from the MA polynomial $$\theta(B) = \det(I - D L(B))$$ of its ARIMA form: $\theta$ is divided by its common factors with $\phi(B) = \det(I - F L(B))$ by [[D6:cyclotomic cancellation]], and the model is accepted when the cancelled polynomial $\theta^*$ has no root in $|B| \le 1$ and, for a model with ARIMA components, the AR polynomial of its ARIMA part is stationary; its largest mode modulus is $1/\min|B|$ over the roots of $\theta^*$. In the numpy prototype (admissibility.py, confirm.py, ets_arima.py) the roots come from numpy; in smooth the [[D7:step-down test]] decides and bisection gives the modulus.
D11 [method] bisection for the largest root modulus: Bisection on $r$ over the [[D7:step-down test]] applied to $\theta^*(B/r)$, which passes exactly when $1/|B| < r$ for every root $B$ of $\theta^*$, so that the smallest such $r$ is the largest mode modulus; run to a width of 1e-10 in the prototype and to a tolerance of 1e-6 in smooth. [part of D10]
D17 [data] ETS(A,N,A) grid: The 3042 [[D9:ETS(A,N,A)]] models with $m = 4$ and $m = 12$ and $\alpha$ and $\gamma$ on the grid 0.05, 0.10, ..., 1.95 (confirm.py, plan PL1).
D18 [data] random GUM models with lags (1, 12): 200 [[D8:GUM]] models with lags (1, 12): entries of $F$ normal with mean 0 and standard deviation 0.35, of $g$ with mean 0 and standard deviation 0.175, of $w$ with mean 1 and standard deviation 0.175 (confirm.py, seed 2026).
D19 [data] random additive ETS and GUM models: The 500 models of plan PL3 (confirm.py --test cancel --seed 2026 --n 200): 200 additive ETS models (level; no, additive or damped trend with $\phi$ in (0.8, 0.99); 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; a constant with probability 0.3; $\beta = 0$ with probability 0.1 and each $\gamma = 0$ with probability 0.15; $\alpha$ in (0.01, 1.9)), 100 GUM models with $F = I$, lags from {1, 4, 12}, $g$ uniform on (-0.3, 0.6) and $w$ normal with mean 1 and standard deviation 0.3, and 200 GUM models with lags from {1, 4, 12, 24} and normal $F$ (standard deviation 0.35). [part of D20]
D20 [data] random additive ETS and GUM models with long seasonal lags: The 575 models of plan PL4 (confirm.py --test stepdown --seed 2026 --n 200): the 500 models of plan PL3 generated afresh, 50 additive ETS models with seasonal lags from {24, 168, 365} and 25 GUM models with lags (1, 24, 168) and normal $F$ (standard deviation 0.3). The generator allows MA polynomial degrees up to 536 (level, trend, seasonal lags 168 and 365, constant).
D24 [data] ETS(A,N,A,A) with seasonal lags 24 and 168: The additive ETS model with a level and two seasonal states of lags 24 and 168, $\alpha = 0.3$ and $\gamma = (0.1, 0.05)$, in the lagged form.
D25 [data] starting point of gum(): The default starting point of gum() with orders (2, 2) and lags (1, 12): $F = I$, $g = 0.1$, $w = 1$.
D26 [design] comparison with a reference: Each model's admissibility decision, and its largest mode modulus where both find it inadmissible, computed by the method studied and by a reference on the same model (the [[D5:PBH reference]], the roots by numpy, the numpy prototype of the check of the whole model or an exact oracle); the result is the number of models whose decisions differ and the largest absolute difference of the modulus. In the planned tests, models whose reference modulus lies within 1e-8 of 1 are left out and counted.
D29 [variable] admissibility decision: Whether a method accepts a model as admissible ([[D3:decaying weights]]); compared between a method and a reference as the number of models on which the two differ.
D30 [variable] largest mode modulus: The largest modulus of the modes that decide admissibility: for the [[D5:PBH reference]] the largest $|\lambda|$ of the reachable and observable eigenvalues of the [[D2:expanded discount matrix]], for the check of the whole model $1/\min|B|$ over the roots $B$ of the cancelled MA polynomial, for the per-lag check the largest $|\lambda|$ of its blocks; a model is admissible when it is below 1.
D41 [method] exact oracle: Admissibility decided in exact rational arithmetic with certified root enclosures (independent/oracle.py, python-flint 0.9.0), written by the contributor P5 from the definitions of this claimset without the prototype: each float of $F$, $g$, $w$ is converted exactly to a rational, $D = F - g w'$, and $$\theta(B) = \det(I - D L(B)), \qquad \phi(B) = \det(I - F L(B))$$ by fraction-free elimination over $\mathbb{Q}[B]$; $G = \gcd(\theta, \phi)$ exactly and $\theta_G = \theta / G$; $C$ is the product of the cyclotomic polynomials $\Phi_n$, $n$ dividing a lag, at the multiplicity dividing both, and $\theta_C = \theta / C$; the roots of $\theta_G$, $\theta_C$ and $G / C$ are enclosed in Arb balls of 128 bits, raised to 512 and 2048 bits while a modulus ball contains 1, and a root counts as on the unit circle only when it is proved a root of unity whose $\Phi_n$ divides the polynomial. The largest mode modulus is $1/\min|B|$ over the roots; a model is admissible when it is below 1.
D42 [data] models of the planned tests: The models of the planned tests PL1-PL6 and the two examples of C10 and C11, drawn again by export_models.py exactly as confirm.py and cpp_check.py draw them (the counts of the original tests are reproduced), with the prototype's values for each: 3242 models of PL1 and PL2, 500 of PL3, 575 of PL4 and PL5 (MA degree up to 535), 800 of PL6 and 2 examples; 5119 in all.
D43 [design] comparison with the exact oracle: For each model, the admissibility decision and largest mode modulus of a method of the study against those of the [[D41:exact oracle]]: the PBH reference against $\theta_G$; the prototype's cancellation, step-down test and bisection against $\theta_C$. Counted are the models whose decisions differ and the largest absolute difference of the modulus on the models both find inadmissible (for the bisection, on every model with a value). Models the oracle cannot decide, and models whose certified largest modulus lies within $10^{-8}$ of 1, are left out and counted.
D44 [variable] non-cyclotomic shared factor in the closed unit disc: Whether $G / C$, the factor shared by $\theta$ and $\phi$ after removing the cyclotomic polynomials $\Phi_n$ ($n$ dividing a lag), has a root in $|B| \le 1$, certified by the [[D41:exact oracle]]; counted over models.

CLAIMS
C3 [methodological; principal; support: simulation E3 (E3.R1, E3.R2) artefacts A1, A3; assumes C14; uses C6, C8, C17; limitations L2, L3; strength moderate (author); verifiability open; planning confirmatory; outcome supported]
  On [[D19:500 random additive ETS and GUM models]], the [[D29:admissibility decision]] from [[D10:the roots of the MA polynomial after cyclotomic cancellation]] differed from the [[D5:PBH reference]] on 0 models, and on the 172 models that both found inadmissible the [[D30:largest mode modulus]] differed from the reference by at most 4.4e-13.
  scope: population: pure additive lagged SSOE models | conditions: additive ETS: level; no, additive or damped trend; 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; optional constant; some persistence fixed at 0; GUM with F = I and lags from {1, 4, 12}; GUM with random F and lags from {1, 4, 12, 24}; numpy prototype; models within 1e-8 of the unit circle excluded (none occurred) | excludes: ARIMA components; CES; adaptive regressors; multiplicative models; seasonal harmonics with non-integer periods (TBATS); the largest mode modulus of admissible models, where a common factor that is not cyclotomic can remain in the cancelled MA polynomial; an eigenvalue of modulus above 1 shared by a reachable and observable mode and an unreachable or unobservable one
  extends B1, at Chapter 11: Hyndman, Koehler, Ord and Snyder relate the reduced forms of linear innovations state space models to ARIMA models.
  test: deterministic; seed 2026; edge_band 1e-08; pbh_tolerance 1e-07; cancellation_tolerance 1e-09; sample 500 models, fixed; no interim analysis; supported if mism == 0 and diff < 0.000001; not supported if mism > 0 or diff >= 0.000001; verdict computed
  plan: lab plan PL3; planned 2026-10-21T15:21:36Z; anchor git_remote
C6 [formal; supporting; extension; support: formal_proof E6, consensus, simulation E10 artefacts A1, A2; verification agent_checked; verifiability open]
  For a [[D1:lagged SSOE model]], ignoring the initial states, $$\det(I - D L(B))\, \epsilon_t = \det(I - F L(B))\, y_t,$$ so the model has the ARIMA form with AR polynomial $\phi(B) = \det(I - F L(B))$ and MA polynomial $\theta(B) = \det(I - D L(B))$.
  scope: population: pure additive lagged SSOE models | conditions: identity of lag-operator polynomials, initial states ignored
  extends B1, at Chapter 11: Hyndman, Koehler, Ord and Snyder relate the reduced forms of linear innovations state space models to ARIMA models.
C8 [formal; supporting; support: formal_proof E8; uses C15; verification agent_checked; verifiability open]
  An eigenvalue $\lambda \neq 0$ of the expanded transition matrix $\tilde F$ with an eigenvector unobservable through $\tilde w$ or a left eigenvector unreachable from $\tilde g$ is an eigenvalue of the [[D2:expanded discount matrix]], and $1/\lambda$ is a common root of $\phi(B) = \det(I - F L(B))$ and $\theta(B) = \det(I - D L(B))$.
  scope: population: pure additive lagged SSOE models
C14 [methodological; supporting; assumption (methodological, tested); verifiability not_applicable]
  The [[D5:PBH reference]] with relative rank tolerance 1e-7 classifies every eigenvalue of modulus at least 1 of the [[D2:expanded discount matrix]] of the tested models correctly as both reachable and observable or not, outside the band of 1e-8 around the unit circle that the tests leave out.
C15 [formal; supporting; support: formal_proof E11; verification agent_checked; verifiability open]
  For the lag-1 expansion $\tilde M$ of $v_t = M v_{t-l}$ in a [[D1:lagged SSOE model]], $$\det(\lambda I - \tilde M) = \det(\Lambda(\lambda) - M),$$ which equals $$\lambda^{\sum_j l_j} \det(I - M L(1/\lambda))$$ for $\lambda \neq 0$.
  scope: population: pure additive lagged SSOE models | conditions: any square matrix $M$ of size $k$
C17 [formal; supporting; support: formal_proof E14, consensus; uses C6, C15; verification agent_checked; verifiability open]
  For a [[D1:lagged SSOE model]] with $\theta(B) = \det(I - D L(B))$ and $\phi(B) = \det(I - F L(B))$, the roots of $\theta(B) / \gcd(\theta(B), \phi(B))$ are exactly the reciprocals $1/\lambda$ of the nonzero eigenvalues $\lambda$ of the reachable and observable part of the [[D2:expanded discount matrix]] (its Kalman decomposition), with their multiplicities.
  scope: population: pure additive lagged SSOE models | conditions: the definition D3 agrees with this part unless an eigenvalue is shared by the reachable and observable part and the rest

EVIDENCE
E3 [simulation; runs 1] Planned test of the cyclotomic cancellation (lab plan PL3, run RN9): confirm.py --test cancel --seed 2026 --n 200 generated 200 additive ETS models (level; no, additive or damped trend with phi in (0.8, 0.99); 0 to 2 seasonal lags from {4, 7, 12, 24, 52}; a constant with probability 0.3; beta = 0 with probability 0.1 and each gamma = 0 with probability 0.15; alpha in (0.01, 1.9)), 100 GUM models with F = I, lags from {1, 4, 12}, g uniform on (-0.3, 0.6) and w normal with mean 1 and standard deviation 0.3, and 200 GUM models with lags from {1, 4, 12, 24} and normal F (standard deviation 0.35); for each it compared the largest root modulus of the cancelled MA polynomial with the PBH reference.
  setup: studied D10; cases D19; procedure D26, D5; lens D29, D30
  E3.R1 admissibility decisions that differ from the PBH reference = 0
  E3.R2 max abs difference of max |lambda| from the PBH reference, inadmissible models = 4.39648e-13 {inadmissible_models: 172}
  E3.R3 models within 1e-8 of the unit circle, excluded = 0
  E3.R4 models compared = 500
  parameters: A3
  seeds: [2026]
E10 [simulation; runs 1] Exploratory run RN2 of explore.py --seed 0: the ARIMA-form polynomials, their cyclotomic cancellation and the PBH reference for ETS(A,N,A,A) with seasonal lags 24 and 168 (alpha = 0.3, gamma = (0.1, 0.05)) and for GUM with orders (2, 2), lags (1, 12), F = I, g = 0.1 and w = 1 (the default starting point of gum()); the determinant identity at random complex B on 100 random GUM models; the expanded discount matrix of a block-diagonal D; and mean times over 50 repeats of three computations for a random GUM with lags (1, 24, 168). The result labelled 'block-diagonal D, lags (1,12)' comes from D = [[0.7, 0], [-0.1, 0.3]], which is lower-triangular rather than block-diagonal.
  setup: studied D1, D6, D10; cases D24, D25; procedure D26, D5; lens D30
  E10.R11 max relative error of the determinant identity = 9.17156e-15 {models: 100}
  E10.R12 max |lambda|, per-lag check = 0.7 {model: block-diagonal D, lags (1,12)}
  E10.R13 max |lambda|, all modes of the expanded discount matrix = 0.904538 {model: block-diagonal D, lags (1,12)}
  E10.R14 mean time, ms = 4.69635 {task: coefficients by DFT, degree: 193}
  E10.R15 mean time, ms = 0.00693922 {task: step-down test, degree: 193}
  E10.R16 mean time, ms = 22.2257 {task: all roots, numpy companion eigenvalues, degree: 193}
  E10.T1 Degree of the MA polynomial theta and of its shared cyclotomic factor, and the largest mode modulus from the roots of theta before and after cancellation and from the PBH reference, for ETS(A,N,A,A) with seasonal lags 24 and 168 and for the default starting point of gum() with orders (2, 2) and lags (1, 12)
  | row | model | degree of theta | degree of the shared cyclotomic factor | max |lambda|, roots of theta before cancellation | max |lambda|, roots of theta after cancellation | max |lambda|, controllable and observable modes (PBH) |
  | r1 | ETS(A,N,A,A), alpha=0.3, gamma=(0.1,0.05) | 193 | 25 | 1 | 0.999997 | 0.999997 |
  | r2 | GUM start, orders (2,2), lags (1,12) | 26 | 14 | 1 | 0.982937 | 0.982937 |
  parameters: A2
  seeds: [0]
E6 [formal; foundation linear algebra over the field of rational functions in $B$] Derivation of the ARIMA form of a lagged SSOE model by the matrix determinant lemma.
  S1 $$\epsilon_t = y_t - w' v_{t-l} \Rightarrow v_t = D v_{t-l} + g y_t,\ D = F - g w'$$ (by D1; substitution)
  S2 $$v_{t-l} = L(B) v_t \Rightarrow (I - D L(B)) v_t = g y_t$$ (by S1, D1; definition of $L(B)$)
  S3 $$\det(I - D L(B))|_{B=0} = 1 \Rightarrow I - D L(B) \text{ is invertible over rational functions}$$ (by S2; evaluation at $B = 0$)
  S4 $$\epsilon_t = \left(1 - w' L(B) (I - D L(B))^{-1} g\right) y_t$$ (by S1, S2, S3; substitution)
  S5 $$I - D L(B) - g\, w' L(B) = I - F L(B)$$ (by D1; definition of $D$)
  S6 $$\det(I - F L(B)) = \det(I - D L(B))\left(1 - w' L(B)(I - D L(B))^{-1} g\right)$$ (by S3, S5; matrix determinant lemma $\det(A - g u') = \det(A)(1 - u' A^{-1} g)$)
  S7 $$\det(I - D L(B))\, \epsilon_t = \det(I - F L(B))\, y_t$$ (by S4, S6; substitution)
E8 [formal; foundation linear algebra over the complex numbers] Unobservable or unreachable modes of the expanded transition matrix are modes of the expanded discount matrix and give common roots of the two ARIMA-form polynomials.
  S1 $$\tilde F u = \lambda u,\ \tilde w' u = 0 \Rightarrow \tilde D u = \tilde F u - \tilde g (\tilde w' u) = \lambda u$$ (by D2; substitution of $\tilde D = \tilde F - \tilde g \tilde w'$)
  S2 $$u' \tilde F = \lambda u',\ u' \tilde g = 0 \Rightarrow u' \tilde D = u' \tilde F - (u' \tilde g) \tilde w' = \lambda u'$$ (by D2; substitution of $\tilde D = \tilde F - \tilde g \tilde w'$)
  S3 $$\det(\lambda I - \tilde M) = \lambda^{\sum_j l_j} \det(I - M L(1/\lambda)),\ \lambda \neq 0$$ (by C15; identity of the lag-1 expansion)
  S4 $$\phi(1/\lambda) = \det(I - F L(1/\lambda)) = 0,\ \theta(1/\lambda) = \det(I - D L(1/\lambda)) = 0$$ (by S1, S2, S3; substitution)
E11 [formal; foundation linear algebra over the complex numbers] Schur-complement derivation of the characteristic polynomial of the lag-1 expansion.
  S1 $$\lambda I - \tilde M = \begin{pmatrix} \lambda I_k - A & -B \\ -C & \lambda I - S \end{pmatrix}$$ (by D2; partition of the states into the $k$ current values (heads) and the $\sum_j (l_j - 1)$ lagged copies (tails); $S$ shifts each tail chain, $C$ feeds each head into its first copy, $A$ and $B$ hold the entries of $M$ that act on heads with lag 1 and on last copies)
  S2 $$\det(\lambda I - S) = \lambda^{\sum_j (l_j - 1)}$$ (by S1; determinant of a lower-triangular matrix with diagonal $\lambda$)
  S3 $$A + B (\lambda I - S)^{-1} C = M\, \mathrm{diag}(\lambda^{1 - l_1}, \dots, \lambda^{1 - l_k}),\ \lambda \neq 0$$ (by S1; $(\lambda I - S)^{-1} C$ maps head $j$ to its copies $\lambda^{-s}$, $s = 1, \dots, l_j - 1$, and $B$ reads copy $l_i - 1$ of component $i$)
  S4 $$\det(\lambda I - \tilde M) = \lambda^{\sum_j (l_j - 1)} \det(\lambda I_k - M\, \mathrm{diag}(\lambda^{1 - l_j})) = \det(\Lambda(\lambda) - M),\ \lambda \neq 0$$ (by S2, S3; Schur complement, then factoring $\lambda\, \Lambda(\lambda)^{-1}$ out on the right)
  S5 $$\det(\lambda I - \tilde M) = \det(\Lambda(\lambda) - M) \text{ for all } \lambda$$ (by S4; two polynomials equal on $\lambda \neq 0$ are equal)
  S6 $$\det(\lambda I - \tilde M) = \lambda^{\sum_j l_j} \det(I - M L(1/\lambda)),\ \lambda \neq 0$$ (by S5, D1; factoring $\Lambda(\lambda)$ out on the right: $\Lambda(\lambda) - M = (I - M \Lambda(\lambda)^{-1}) \Lambda(\lambda)$)
E14 [formal; foundation linear systems theory over the complex numbers] The reduced form of the ARIMA polynomials through the Kalman decomposition of the expanded system.
  S1 $$\exists T: T^{-1}\tilde F T \text{ block-triangular with a reachable and observable block } (F_{co}, g_{co}, w_{co}),\ \tilde w'(zI - \tilde F)^{-1}\tilde g = w_{co}'(zI - F_{co})^{-1} g_{co}$$ (by D2; Kalman canonical decomposition)
  S2 $$\tilde D = \tilde F - \tilde g \tilde w' \text{ keeps the reachable and the unobservable subspaces, so } D_{co} = F_{co} - g_{co} w_{co}'$$ (by S1, D2; reachability is invariant under state feedback and observability under output injection)
  S3 $$1 + w_{co}'(zI - F_{co})^{-1} g_{co} = \frac{\det(zI - D_{co})}{\det(zI - F_{co})},\ \gcd(\det(zI - D_{co}), \det(zI - F_{co})) = 1$$ (by S2; matrix determinant lemma; a single-input single-output realisation is minimal if and only if the numerator and denominator of its transfer function are coprime)
  S4 $$1 + \tilde w'(zI - \tilde F)^{-1}\tilde g = \frac{\det(zI - \tilde D)}{\det(zI - \tilde F)} = \frac{\theta(1/z)}{\phi(1/z)}$$ (by C15, D1; matrix determinant lemma and the identity of the lag-1 expansion)
  S5 $$\frac{\theta(B)}{\gcd(\theta, \phi)}(B) = 0 \iff B = 1/\lambda,\ \lambda \in \mathrm{spec}(D_{co}) \setminus \{0\} \text{ (with multiplicity)}$$ (by S3, S4; uniqueness of the reduced fraction, with z = 1/B: the eigenvalues at zero give no root, theta(0) = phi(0) = 1 makes theta / gcd the reduced numerator, and the reversal keeps coprimality away from B = 0)
E25 [simulation; runs 2] Planned test of the contributor P5 (lab plan AC5.PL1, run AC5.RN8, with the breakdown run AC5.RN9): the [[D41:exact oracle]], written without the prototype, on the [[D42:models of the planned tests]] exported by runs RN57-RN61, against the prototype's values stored with each model. The oracle was tried beforehand only on models of other seeds (AC5.RN2, AC5.RN6, AC5.RN7) and on hand-checkable models. AC5.RN8 took 30 s.
  setup: studied D5, D10, D7, D11, D4; cases D42, D17, D18, D20; procedure D43, D41; lens D29, D30, D44
  E25.T1 Each set of models of the planned tests (D42) given to the exact oracle (D41), and over all sets: models given, left out by the edge band and undecided; decisions of the PBH reference and of the prototype's cancellation that differ from the oracle, and the largest absolute difference of the largest mode modulus on the models both find inadmissible; models with a non-cyclotomic shared factor in the closed unit disc (D44); models whose cancelled MA degree differs from the prototype's. Empty cells: the set has no such value of the prototype (perlag holds the PBH reference only; stepdown and cpp the cancellation only).
  | row | set | models given to the oracle | models on which the oracle failed | models compared | models whose shared non-cyclotomic factor could not be located | models with a shared non-cyclotomic factor, all roots in |B| > 1 | models excluded by edge | models excluded by undecided | admissibility decisions of the PBH reference that differ from the exact oracle | max abs difference of the largest mode modulus, PBH reference against the exact oracle, models both find inadmissible | admissibility decisions of the prototype's cancellation that differ from the exact oracle | max abs difference of the largest mode modulus, prototype's roots after cancellation against the exact oracle, models both find inadmissible | models with a shared factor of theta and phi that is not a product of Phi_n, n dividing a lag, with a root in |B| <= 1 | models whose cancelled MA degree differs from the prototype's |
  | r1 | perlag | 3242 | 0 | 3164 | 0 | 0 | 78 | 0 | 0 | 4.21885e-15 |  |  | 0 |  |
  | r2 | cancel | 500 | 0 | 500 | 0 | 8 | 0 | 0 | 0 | 4.66294e-15 | 0 | 4.39426e-13 | 0 | 0 |
  | r3 | stepdown | 575 | 0 | 575 | 0 | 8 | 0 | 0 |  |  | 0 | 6.58518e-12 | 0 | 0 |
  | r4 | cpp | 800 | 0 | 797 | 0 | 8 | 3 | 0 |  |  | 0 | 5.20488e-10 | 0 | 0 |
  | r5 | examples | 2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
  | r6 | all | 5119 | 0 | 5038 | 0 | 24 | 81 | 0 | 0 | 4.66294e-15 | 0 | 5.20488e-10 | 0 | 0 |
  E25.T2 The 575 models of PL4 and PL5 (MA degree up to 535 before cancellation): step-down decisions of the prototype that differ from the exact oracle; the largest absolute difference of the bisection value from the certified largest mode modulus, over the models that have one; and the degrees. {set: stepdown}
  | row | set | step-down decisions of the prototype that differ from the exact oracle | max abs difference of the largest mode modulus, prototype's bisection against the exact oracle | models with a bisection value | max degree of theta, exact | max degree of the cancelled MA polynomial, exact |
  | r1 | stepdown | 0 | 5.7387e-11 | 198 | 535 | 533 |
  E25.T3 The models of PL1 and PL2 (3042 ETS(A,N,A) grid models, 200 GUM models with lags (1, 12)): the counts of C1 and C2 with the exact oracle in place of the PBH reference and the band of 1e-8. {set: perlag}
  | row | family | models accepted by the per-lag check that do not forget their initial states, exact oracle | models rejected by the per-lag check that forget their initial states, exact oracle | models accepted by the per-lag check | models of the family |
  | r1 | ETS(A,N,A) | 1482 | 0 | 3042 | 3042 |
  | r2 | GUM | 10 | 1 | 193 | 200 |
  E25.T4 The models in the edge band, by set and family (run AC5.RN9): their number, the largest distance of the certified largest mode modulus from 1, those with roots proved on the unit circle, and those whose decision by the method of the original test (the PBH reference for perlag, the prototype's cancellation for cpp) differs from the oracle's, with the largest modulus difference.
  | row | set | family | models in the edge band | max distance of the certified largest mode modulus from 1, edge models | edge models with roots proved on the unit circle | edge models whose decision differs from the exact oracle | max abs difference of the largest mode modulus against the exact oracle, edge models |
  | r1 | perlag | ETS(A,N,A) | 78 | 4.44089e-16 | 2 | 6 | 2.44249e-15 |
  | r2 | cpp | ETS high degree | 3 | 6.89469e-09 | 0 | 0 | 3.70814e-14 |
  E25.T5 The examples of C10 and C11 (run AC5.RN9): the certified largest mode modulus, its difference from the PBH reference and from the prototype's roots after cancellation, and the exact degrees of theta, of the shared cyclotomic factor and of the gcd of theta and phi.
  | row | model | largest mode modulus, exact oracle (theta_G) | abs difference of the largest mode modulus, PBH reference against the exact oracle | abs difference of the largest mode modulus, prototype's roots after cancellation against the exact oracle | degree of theta, exact | degree of the shared cyclotomic factor, exact | degree of the exact gcd of theta and phi |
  | r1 | ETS(A,N,A,A), alpha=0.3, gamma=(0.1,0.05) | 0.999997 | 1.66533e-15 | 1.11022e-16 | 193 | 25 | 25 |
  | r2 | GUM start, orders (2,2), lags (1,12) | 0.982937 | 7.77156e-16 | 2.22045e-16 | 26 | 14 | 14 |
  parameters: A19

LIMITATIONS
L2 [measurement; affects C1, C2, C3, C10, C11, C16; could_reverse; mitigation addressed] The PBH reference and the cyclotomic cancellation were written by the same agent and share the lag-1 expansion code and numpy's LAPACK eigenvalue routines.
L3 [sampling; affects C2, C3, C4, C5, C16; reduces_transferability; mitigation none] The random model sets (persistence ranges, the probabilities of zero persistence and of a constant, the standard deviations of the GUM matrices) were chosen by the agent and not drawn from models estimated on data.

ARTEFACTS
A1 [code] numpy prototype of the admissibility check of pure additive lagged SSOE models: lag-1 expansion, the per-lag check of smoothEigens(), the PBH reference, ARIMA-form polynomials by inverse DFT, cyclotomic cancellation, step-down test and bisection; ETS and GUM model builders
A2 [code] exploratory script: ETS(A,N,A) cases, multiple seasonality and the GUM starting point, determinant identity, step-down and bisection on random GUMs, timings
A3 [code] test script: per-lag check against the PBH reference, cancellation against the PBH reference, step-down decisions and bisection against roots, on seeded random model sets
A18 [data] Every result of the lab record at full precision, one row per result: the lab result, the run that produced it, the quantity, its conditions (JSON) and the value.
A19 [code] The exact oracle of the contributor P5 (with test_oracle.py and, from commit e4963c49, edge_report.py in the same folder): exact theta, phi, gcd and cyclotomic cancellation over the rationals, certified root enclosures with python-flint 0.9.0, and the comparison with the prototype's values.
A20 [code] Draws the models of the planned tests PL1-PL6 as confirm.py and cpp_check.py do and writes them, with the prototype's values for each, to models.json (lab runs RN57-RN61).

REFERENCES
B1 10.1007/978-3-540-71918-2 (doi)
B2 10.1016/0047-259X(73)90030-4 (doi)
B3 10.32614/CRAN.package.smooth (doi)

HOW TO CITE
This claim: Svetunkov, I. (guarantor) (2026). Claim C3, in: Admissibility of lagged state space models through the ARIMA form of the whole model. EntryLib. entrylib:2026.000002@v3#C3. https://entrylib.org/p/2026.000002/C3
The claimset it is from: Svetunkov, I. (guarantor) (2026). Admissibility of lagged state space models through the ARIMA form of the whole model. EntryLib. entrylib:2026.000002@v3. https://entrylib.org/p/2026.000002@v3